Here are the solutions to the trigonometric equations.
: Find the value of θ between 0∘ and 360∘ for the equation sin(2θ)=0.5.
Step 1: Find the reference angle.
Let X=2θ. The equation becomes sin(X)=0.5.
The reference angle α for sin(α)=0.5 is:
α=sin−1(0.5)=30∘
Step 2: Determine the quadrants where sin(X) is positive.
Since sin(X) is positive, X must be in the 1st or 2nd quadrant.
Step 3: Find all possible values for X in the range 0∘≤X≤720∘.
Since 0∘≤θ≤360∘, the range for X=2θ is 0∘≤2θ≤720∘.
In the 1st quadrant:
X=30∘X=30∘+360∘=390∘
In the 2nd quadrant:
X=180∘−30∘=150∘X=150∘+360∘=510∘
So, the values for X are 30∘,150∘,390∘,510∘.
Step 4: Solve for θ.
Substitute X=2θ and divide by 2:
2θ=30∘⟹θ=230∘=15∘2θ=150∘⟹θ=2150∘=75∘2θ=390∘⟹θ=2390∘=195∘2θ=510∘⟹θ=2510∘=255∘
All these values are within the range 0∘≤θ≤360∘.
The values of θ are 15∘,75∘,195∘,255∘.
: Find the value of θ to the nearest degree for cos(3θ+20∘)=23 where 0∘≤θ≤360∘.
Step 1: Find the reference angle.
Let Y=3θ+20∘. The equation becomes cos(Y)=23.
The reference angle α for cos(α)=23 is:
α=cos−1(23)=30∘
Step 2: Determine the quadrants where cos(Y) is positive.
Since cos(Y) is positive, Y must be in the 1st or 4th quadrant.
Step 3: Determine the range for Y.
Given 0∘≤θ≤360∘:
Multiply by 3: 0∘≤3θ≤1080∘
Add 20∘: 0∘+20∘≤3θ+20∘≤1080∘+20∘
So, 20∘≤Y≤1100∘.
Step 4: Find all possible values for Y in the range 20∘≤Y≤1100∘.
In the 1st quadrant:
Y=30∘Y=30∘+360∘=390∘Y=30∘+720∘=750∘
In the 4th quadrant:
Y=360∘−30∘=330∘Y=330∘+360∘=690∘Y=330∘+720∘=1050∘
The next values (30∘+1080∘=1110∘ and 330∘+1080∘=1410∘) are outside the range Y≤1100∘.
So, the values for Y are 30∘,330∘,390∘,690∘,750∘,1050∘.
Step 5: Solve for θ and round to the nearest degree.
Substitute Y=3θ+20∘:
3θ+20∘=30∘⟹3θ=10∘⟹θ=310∘≈3.33∘≈3∘3θ+20∘=330∘⟹3θ=310∘⟹θ=3310∘≈103.33∘≈103∘3θ+20∘=390∘⟹3θ=370∘⟹θ=3370∘≈123.33∘≈123∘3θ+20∘=690∘⟹3θ=670∘⟹θ=3670∘≈223.33∘≈223∘3θ+20∘=750∘⟹3θ=730∘⟹θ=3730∘≈243.33∘≈243∘
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ScanSolve AI Answer
Here are the solutions to the trigonometric equations. Problem 1: Find the value of between 0^ and 360^ for the equation (2) = 0.5. Step 1: Find the reference angle. Let X = 2. The equation becomes (X) = 0.5. The reference angle for () = 0.5 is: = ^-1(0.5) = 30^ Step 2: Determine the quadrants where (X) is positive. Since (X) is positive, X must be in the 1st or 2nd quadrant. Step 3: Find all possible values for X in the range 0^ X 720^. Since 0^ 360^, the range for X = 2 is 0^ 2 720^. In the 1st quadrant: X = 30^ X = 30^ + 360^ = 390^ In the 2nd quadrant: X = 180^ - 30^ = 150^ X = 150^ + 360^ = 510^ So, the values for X are 30^, 150^, 390^, 510^. Step 4: Solve for . Substitute X = 2 and divide by 2: 2 = 30^ = (30^)/(2) = 15^ 2 = 150^ = (150^)/(2) = 75^ 2 = 390^ = (390^)/(2) = 195^ 2 = 510^ = (510^)/(2) = 255^ All these values are within the range 0^ 360^. The values of are 15^, 75^, 195^, 255^. Problem 2: Find the value of to the nearest degree for (3 + 20^) = sqrt(3)2 where 0^ 360^. Step 1: Find the reference angle. Let Y = 3 + 20^. The equation becomes (Y) = sqrt(3)2. The reference angle for () = sqrt(3)2 is: = ^-1(sqrt(3)2) = 30^ Step 2: Determine the quadrants where (Y) is positive. Since (Y) is positive, Y must be in the 1st or 4th quadrant. Step 3: Determine the range for Y. Given 0^ 360^: Multiply by 3: 0^ 3 1080^ Add 20^: 0^ + 20^ 3 + 20^ 1080^ + 20^ So, 20^ Y 1100^. Step 4: Find all possible values for Y in the range 20^ Y 1100^. In the 1st quadrant: Y = 30^ Y = 30^ + 360^ = 390^ Y = 30^ + 720^ = 750^ In the 4th quadrant: Y = 360^ - 30^ = 330^ Y = 330^ + 360^ = 690^ Y = 330^ + 720^ = 1050^ The next values (30^ + 1080^ = 1110^ and 330^ + 1080^ = 1410^) are outside the range Y 1100^. So, the values for Y are 30^, 330^, 390^, 690^, 750^, 1050^. Step 5: Solve for and round to the nearest degree. Substitute Y = 3 + 20^: 3 + 20^ = 30^ 3 = 10^ = (10^)/(3) ≈ 3.33^ ≈ 3^ 3 + 20^ = 330^ 3 = 310^ = (310^)/(3) ≈ 103.33^ ≈ 103^ 3 + 20^ = 390^ 3 = 370^ = (370^)/(3) ≈ 123.33^ ≈ 123^ 3 + 20^ = 690^ 3 = 670^ = (670^)/(3) ≈ 223.33^ ≈ 223^ 3 + 20^ = 750^ 3 = 730^ = (730^)/(3) ≈ 243.33^ ≈ 243^ 3 + 20^ = 1050^ 3 = 1030^ = 1030^
